globally polar sets for plurisubharmonic functions

(1)

globally polar sets for plurisubharmonic functions

o n

C n

Bengt Josefson*

We shall prove that a locally polar set in C" is globally polar which generalizes a well-known result from potential theory for subharmonic functions and answers a question posed by Lelong [2]. Our method differs from the ones frequently used in potential theory, since it seems that there is a lack in the representation o f plurisubharmonic functions by kernels, and the main step in our p r o o f is to find, to every given function which is analytic in a ball, polynomials which are sufficiently small on the set where the given function is small (Proposition). F r o m this the theorem will follow (Lemma 3) because locally a plurisubharmonic function is a Hartogs function. A consequence of the theorem is that an analytic set is globally polar and the theorem also has applications in the theory for capacities and extremal functions in C ". See for example Siciak [3].

Definition. A set D e C " is called locally polar if there exist, to every zED, an open set V z c C ~ and uzEPSH(V~), where PSH(Vz) denotes the set o f all plurisubharmonic functions in Vz, so that zEV~ and such that u J V z n D , the restriction ofu~ to V~nD, is equal to - oo. D is globally polar if we can take Vz~-C ".

F o r details see [2].

We shall give C ~ the sup-norm and we shall let ~ ( V ) , where V c C ~ is open, denote the set o f all analytic functions on V. We note that f has a Taylor series expansion f ( z ) = ~ a , z " if f E g ( B ( 0 , S)), where B(0, S) is the open bali in C ~ with centre 0 and radius S, a, EC, r=(rl .... , r~) is a multi-index and zr=z~ I... z r.

where z = (z 1 . . . z,) E C ~.

Theorem. A set D c C ~ is globally polar if and only if D is locally polar.

From the theorem we obviously have the following,

* Supported by the Swedish Natural Science Research Council Contract No. F 3435---004.

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110 Bengt Josefson

Corollary.

An analytic subset of an open set in C" is globally polar.

We note that the "only i f " part of the theorem is evident. For the rest o f the proof we need a number of lemmas.

Let D ~ C " be a locally polar set. From the definition it follows that, for every

zED,

there exist r z > 0 and

uz~PSH(B(z, rz))

such that

uJB(z,

r,)c~D . . . .

Let from now on z be fixed. We shall first show that

B(z, rJ32)c~D

is a globally polar set. Without loss of generality we may assume that z = 0 and r0=4. To avoid too many subscripts we shall write u instead of u0 and it is obvious that we can take u such that u ( z ) < 0 when Ilzll-<_2.

From Bremermann [1] we easily get the following:

Lemma 1.

We can write u(z)= lI'i-mz,_,l~j_=(1/j) log

Ifi(z')I

where fj(z)6 ~(B(0, 4)) and [IfyllB<0,2)=

^sup

Ifj(z)[ <- 1.

ELzlI~2

Proof.

From [1] it follows that

n = {(z, w)~Cn+t;

z~B(O,

4) and Iwl < e -~z)}

is an open pseudoconvex set. Since u < 0 when Ilzll<_-2 we have that K =

{(z, w); Ilzll<=2

and

Iwl~_l}

is a compact subset of H. The theorem of Bremer- m a n n - - N o r g u e t - - O k a gives that there exists an

f~Yg(H)

which cannot be continued over H and so that

I[flIK=sup~z.~)~KIf(z,w)t<I.

We can write

f(z, w)=~wJf~(z)

where f j ~ e ( B ( 0 , 4)) and

u(z)

= ~ l-~ (I/j)log If~(z')l

according to [1]. Since Ilfll~c<l it follows that IIf~llB~o,2)<l which completes the proof. Q.E.D.

There exists an integer q > 0 such that sup~E~/4

u ( z ) > - q + l .

Hence there exists an infinite set S c N so that

IlfjllB~o.a/4)>e-qJ

when

j~S.

Since

l-~,_~l-~jEs(l/j)log tf~(z')t<-_u(z)

we may assume that equality holds, i.e. u is defined by (fJ)jcs. We m a y also assume that (nj)2~<2 J when

jES.

Next we will find, to e v e r y f j , a polynomial gj of degree

ij

such that [gj(z)l~/iJ is small when

Ifi(z)] x/j

is small. We cannot expect the Taylor series to give such a good approximation in the set where (f~)x/J is small or such a good approximation for example on a ball and have to find other methods.

Put

N(s)={fE~/g(B(O,

3)); Ilfllsc0,x)<=l and ]f(0)[>e-~}. We note that there exists, for every

j~S, xJCB(O,

1/4) such that

f~'(z)=f~(z-x~)~N(qj)

since f / ~ g ( B ( 0 , 4)),

IIf~ll~(o,2)<=l

and

IIf~ll~(o, xz~)>e-~L

Proposition.

Let f~ N(j), where

j ~ R +

is so big that (nj)~<2 j, and let

r

Then there exists a polynomial g such that

l~_[Igllnr ~,

where i is the degree

(3)

of g, and so that Ig(z)l<exp(-ciq~w) when [f(z)[<exp(-j~o) and

[Izll~l/2,

where C=l/2.10an.

Proof.

First we note that it is no restriction to assume that q~l/~ is an integer, because if the proposition is true for every such tp with C = 1/(10an) (as we shall prove), then it holds for every q~>100 with C as in the proposition, since then [tpl/~]--I > 1 , where [ ] denotes the integer part. It is also easy to see that we may suppose that j is an integer.

Furthermore, we may assume that ~o<-j since we can always raise f to the power q~ and if g exists relative to

P'EN(jq~)

as in the proposition, g also has the desired properties relative to f.

Let

f ( z ) = ~ a , z "

and let M c N " be the set

M={r; r~<jq~}.

It is clear that M contains exactly

jnq~n

different elements. Put

Q(z)=~,~Mxrz"

and H ( z ) =

f ( z ) Q ( z ) = ~ d , z ' ,

where

d,=~t~Ma,_tx ,

where we put a , _ , = 0 if min, (r~- t~)<0.

Now (d,=0),EM is a system of linear equations in the variables

x t

and with coefficients a,. There are j~ tp ~ variables and equations. Let D (M) be the determinant o f the system.

We note that

H(z)

is small when

f(z)

is small since H is a product of a polynomial and f . We shall show that x t can be chosen so that dr = 0 when [[rll = ~'~ r~-<_

j , q~/2 and max

r~>A =i/n=

100jq~ (n-wn and so that at least one d,, with Ilrll <=i, is big (at least bigger than e-J'o/l~ Then it will follow that G(z)=~'m~x, ~a

d,z r

is small when

f(z)

is small, since G is almost H, and that G has the desired properties, i.e. G is not small on the unit ball. That the variables x t can be taken in the way described above follows from the fact that if all d, are small when ILrI[ ~_i then the system of equations {d,=0}r~M can be slightly changed so that the new system has a non-trivial solution, thus the determinant of the new system is zero since the system has as many variables as equations. But then it follows that there exists a submatrix o f {dr=0}reM with a determinant which is much bigger than that of {dr=0 } and since

D(M)

is big, a repetition o f this argument will lead to a contradiction because la, I <-- 1.

We shall first show that

D(M)=(f(O)) J"~'".

This follows from the fact that the coefficient for x t in d t is a 0 = f ( 0 ) and because the coefficient for x t in d r is 0 if

r~<t,

for some sE(1 . . . . ,n). Hence the matrix belonging to the system (dr=0)reM is zero on one side o f the diagonal and with diagonal dements equal to f ( 0 ) which gives that

D(M)=(f(O)) i~'".

Let N and

N ' c M

be such that z ( N ) = T ( N ' ) , where z denotes the number o f elements. Let {d(N')=0}r~N be the system o f linear equations

~teN, ar-txt=O,

tEN

and let

D(N, N')

denote its determinant which exists since z ( N ) = ~ ( N ' ) . We have that

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112 Bengt Josefson

(1)

ID(N,N')I<(j"q~")J"~'"<e i"+'"*

if jU<--e / since q~<_--j, the number of equations in the system is less or equal to (j~o)" and since

la, l<=l

^because

fEN(j).

Let

M k

and

N k C M be

such that a) z(Mk) = x(Nk) = x ( M ) - - k =j"~o"-k,

b) rC M, if rE M and max s rs > 100j~0 (~-1)/" = A , c)

ID(Mk,

NDI > exp

(kjtp/lO-j"+ltf).

Mo=No=M

fulfil the requirements, since

ID(M,M)I=ID(M)f=If(O)IJ"~">

e -j"§ (Since

f~N(j)).

According to (1) there exists a biggest integer m so that M m and N m exist and satisfy the conditions a)---c). We also have from (1) that

(2) rn < 20j"q~ "-1 .

There exists

r~

such that

max~r~ ("-x)/".

This follows because there are

(A+l)">lOOj"qf-l>m

different

rEM

with maxsr~-<-A, since

A<flp

if q~ > 100.

Put M,~+l=Mm\{r~ The system of linear equations Z~t~N.. a,-txt = O. r~ Mm+ 1

has a nontrivial solution, since the number of variables

xt

is x (Nm)=z ( M ) - - m and the number of equations is

x ( M m + a ) = x ( M ) - m - 1 .

Let {ut} be a solution such that maxt [utl=l and take

t~

so that

[u:l=l.

We shall now prove that

(3)

In,0[ =

[Z,,N~, a:-,u,I ~-

e-J*ll~

Put

b:,,o:a,o_,o--(~t~N,a,o_,ut)/uto

and

b,,t=a,_ t

when

r # r ~

or

t ~ t ~

We have

7~t~N,. b:,tut=a,o-toUto--~tE~r a,o-tUt+~tEN,.,t,,toa,o-tut=O

and

,~t~N,, b,,tut=~tCN., a,-tut =0,

when

r~ Mm+ a

according to the choice o f

{u,}.

Thus the system o f linear equations

z~t~N,b,,txt=O, r 6 M m

has the nontrivial solution {ut}, hence the determinant D of the system, which exists since the number of variables is equal to the number of equations (z(Mm)=z(Nm)), is zero. Put

Nm+a=Nm\{t~

Then

D = D ( M m, Nm)+(b,o, to-a,o_to)" D(Mm+l,

Nm+l)=0 since

b,a=a,_ t

when

r r ~

or

t ~ t ~

Trivially it follows that

a)

"r(nm+x)=z(Nm+t)=z(M)--m-- 1

b)

rEMm+ x

if

rEM

and max, r~>A, because maxsr~ and

Mm=Mm+tw{r~

Because of the choice o f m (m is the biggest integer so that a)--c) are fulfilled for any sets Mm and

N , , c M ) ,

we must have that

[D(Mm+,,Nm+~)I=

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[b,o,o-a,o-tol -a iD(Mm, Nm)l

<-exp ( ( m +

1)j~o/lO-j"+lq) ")

hence that Ib,o.to-a,o ,o1 e -j*/l~ because of c). But

Ib,o,o--a,o_,ol=lY,,~Na,o_,U,i/lu, ol=Ia,ol,

since lu, ol=l, and thus (3) is established.

We shall now proceed to construct the polynomial g in the proposition.

Let H(z), L)(z) (resp. a,) be the functions (resp. complex numbers) which are obtained from

H(z), Q (z)

(resp. d,) when we replace the complex variables {xt}

by the complex numbers

{ut}.

Then 10(z)l<-(jq~)" when Ilzll<_-I since [u,l<-l.

Hence

IH(z)[<(jq))"e --J~'

if

If(z)l<e-J~"

and llzll~l.

Put

G(z)=~maxsr~_.tdrz r

and I[rl[ = z ~ ' r s . Then

H ( z ) - G ( z ) = z~,,,>j~,12d, z',

because

[[rll<Jq)/2

when maxsrs<_-A and (p>100, and because d , = 0 if

[Irll<=jg/2

and maxr~>A. The last assertion follows from the fact that

rEM

if IIr[I

<=j9/2,

hence that

rEMm

and also

rEM,~+~

according to b), if max

r~>A,

and from the fact that fit=0 when

rEMm+x

(the construction of {ut}). For every r we also have that 13,l<=(j~o)" since

lu, l<=l

and [a,[<=l

(fEN(j)).

Thus

IR(z)-G(z)i<-~lt,ll>je/2(jq))n2 -It'tl

if [Izll<=l/2 since we have given C n the sup- norm. But s ' 9 - ^{II rll}~_. " ~ In ,')--l.... ,')-- jq~/2 a j o / 5 since 100-<_ (p <=j and

j2"<eJ<dq'/5.

Hence IH(z)-G(z)l<=e -j~'/4 and so IG(z)l<e -j~'/5 when Ilzll<=l/2 and

if(z)l<e -j~'

since then, according to the above,

[B(z)]<(jq))~e-J~'<e -je+j/5.

Put d=max,~_a

la, I.

We have that

d>e -j~'/l~

since

la,01>e -j~/~~

according to (3) (maxr~ Finally put

g(z)=d-lG(z).

Then

gEPi(C n)

where

i=An,

since GEPi(C0. It is also true that 1 <-11 glJn(0,~) <=2i because m a x , . , , < a d - l l a , l = l and because

(nA)n<2 i

(Since ( j n ) ~ < 2 0 . We have further that

Ig(z) I -<_

e i~'ll~ = e -jell~

= exp

(iq)llnlnl03)

when If(z)l <- e -j*

and llzll<=l/2, since

d-~<-e x~'/l~

and

iG(z)i<e-J~'15

in that case. Thus g has the properties in the proposition which completes the proof. Q.E.D.

Proof of the theorem continued.

Take, for every f~ (defined as before the Pro- position) and every integer r=>10,

i(j,

r)EN and gj,,EPi(s,o(C n) as in the Proposi- tion such that

(1) lgj,,(z)l <

e x p ( - c i ( j , r ) r ~)

when

Ifj(z)l

<

exp(-jqr 2")

and Ilzll ~- 1/2.

Put t j - / - / , = l 0 i(j, r) and - J

ej,,(z)=(g~,,(z+x~))'t i~i'O.

We note that

(2) 2 -tj -~

llej,,llB(o,1)

~- 4b

since sup

[ej,,(z-M)[

= sup

Igj.,(z)l'/"s,'~ >=

tj-l(3/g)tJ > 2-t~

II z II ~-- 3 / 4 I1 z n ~- 3/0,

and since

sup les.,(z-x~)l ^<-

ts(514)b2'~ < 4b because 1 <- Ilgj,,]l~(o,x) ~- 2i~

II z U --~ 514

We also note that

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114 Bengt Josefson

(1)"

lei.,(z)l<exp(-Ctjr ~')

when

If~(z)l<exp(-jqr ~"')

follows from (1).

Put

h~(z)=Hxo~_,~_j (e~.,(z)) "-~

and finally

v(z) = I ~ ~ (1/tj) log

[h~(z')l

z ' ~ z j ~ S , j ~

and

I/zll

<- 1/4 which

where S is as before the Proposition.

L e m m a 2 .

vEPSH(C ~) and

v ( z ) = - o o /f

zEDc~B(O,

1/8).

Proof.

We have that

v(z)<8k

when I l z l l ~ k ~ l because (2) gives that

I[ej,,lla~O,k)~tjld~4'J<(8k)t~

and because ~,_~a0 r - ' < 1.

Put

D~.,={zEB(O,

1);

]ej.,(z)ll/e"~

1--2-'} and let

L(Dj,,)

be the Lebesgue measure of Dj, r.

Because of (2) there exists f i " ~ B ( 0 , 1) such that

lej,,(yJ")[1/"'J~2 -~-"

and since log

le~,~(yJ"+z)l

is plurisubharmonic we then have that

1

f,

~, ~_, (1/? ti) log

I%.(YJ'

~ + z)!

dz ~ - r -r

log 2.

(4n) ~

Furthermore, since Ile~,,llB~0.a)~24'~ according to the above we have that 1

f,,,,,

~_~ (1/?t~) log

[ej,,(y j," + z)[ dz ~ r-"

log 24 +L(Dj.,) log ( 1 - 2 - ' ) . (4n)"

Together with the inequality above this gives that

L(Dj,,) <=

r - ' l o g 4 8 / l o g ( 1 - 2 - ' ) < r-'+12 "+1 < 2-" if r ~ 10.

Thus it follows from the construction of hj that Ilhjlln~o,l~>2-tJ since B(0, 1)\U,~10Dj, r is not empty because /r/,_~10(1-2-')>1/2. Hence [1] or Hartogs' Lemma gives that v ~ - ~ , that is,

vEPSH(C~).

We shall now show that v ( z ) = - - ~ when

zCDc~B(O,

1/8). Assume that this is not true. Then there exist

z~DnB(O,

1/8) and a constant - ~ < T < 0 such that

v ( z ) > T + l .

Hence there exist, for every m~N, a vector

z~'CB(O,

1/4)and an infinite set S ~ S such that

z " ~ z

as rn~oo and so that

Ihj(z")l~ertj

when

jES,,.

Take

IEN

so big that

- - I t C < T - 2

where C is defined in the Proposition.

According to (2),

[ej,~(z")l<ee'~

and hence

lI~o~,~j,..lej,.(zm)l~-'<e~'~.

^But

thj(z")l>-ertJ

when

j 6 S m

so it follows that

lej,,(z~")l>=exp((T-2)tfl')>

exp

(--Ctfll).

Thus (1) gives that [fj(z')] =>exp

(-jql 2~z)

when

j~ Sm.

That implies that

u ( z ) > - q l 2~l

since u ( z ) = l ~ , , _ lli-mj_~ (1/j)log

Ifj(z')l

which contradicts the fact that

zEDc~B(O,

1/8) and completes the proof. Q.E.D.

(7)

We have proved that D caB(0, 1/8) is globally polar hence that D c~B(z, r~/a2 ) is globally polar. Since z is arbitrarily taken in D it is enough to prove the following lemma to complete the p r o o f o f the theorem.

Lemma 3. I f there exists, to every zED, a ball B(z, G) such that DcaB(z, r,) is globally polar then D is globally polar.

Proof Obviously UzcDB(z, r~) is open and hence o--compact. Thus there exist countably many zJED such that D c B ( z j, rzj). But it is well known and easily seen that a countable union o f globally polar sets is a globally polar set, which proves the Lemma and thus completes the p r o o f of the Theorem. Q.E.D.

References

I. BREMERMANN, H. J., On the conjecture of the equivalence of the plurisubharmonic functions and the Hartogs functions, Math. Ann. 131 (1956), 76---86.

2. LELONO, P., Ensembles singuliers impropres des fonctions plurisousharmoniques, J. Math.

Pures AppL 36 (1957), 263--303.

3. SICIAK, J., Extremal plurisubharmonic functions in C N, (to appear).

Received September 23, 1976 Bengt Josefson

Department of Mathematics Uppsala University Thunbergsv/igen 3 752 38 UPPSALA Sweden

globally polar sets for plurisubharmonic functions